Residual Arithmetic, r=3, d=10

The following table illustrates the performance of residual arithmetic.
The dimension of the spline space on the generic double Clough-Tocher
split, for r=3 and d=10 is 184. It was computed
using three consecutive prime numbers. The table gives the top one of
those prime numbers, and the color indicates the result, as follows:

Red:

All
entries in the linear system have at least one of their residuals
equal to zero. This makes Gaussian Elimination impossible and the
matrix is considered to have rank 0.

Purple:

The computed dimension is too high. Some non-zero numbers in the linear
system are treated as being zero.

Cyan:

The dimension is computed correctly. However, some entries in the
linear systems have a mixture of zero and non-zero residuals. The
program recognizes this as a potential problem. Numbers with mixed
residuals are considered non-zero, but they cannot serve as pivots.

Green:

The dimension is computed correctly and all non-zero entries in the linear system have three non-zero residuals. This is the expected and desired situation.
The smallest triple of primes having these properties are 2423, 2437, 2441.

5

7

11

13

17

19

23

29

31

37

41

43

47

53

59

61

67

71

73

79

83

89

97

101

103

107

109

113

127

131

137

139

149

151

157

163

167

173

179

181

191

193

197

199

211

223

227

229

233

239

241

251

257

263

269

271

277

281

283

293

307

311

313

317

331

337

347

349

353

359

367

373

379

383

389

397

401

409

419

421

431

433

439

443

449

457

461

463

467

479

487

491

499

503

509

521

523

541

547

557

563

569

571

577

587

593

599

601

607

613

617

619

631

641

643

647

653

659

661

673

677

683

691

701

709

719

727

733

739

743

751

757

761

769

773

787

797

809

811

821

823

827

829

839

853

857

859

863

877

881

883

887

907

911

919

929

937

941

947

953

967

971

977

983

991

997

1009

1013

1019

1021

1031

1033

1039

1049

1051

1061

1063

1069

1087

1091

1093

1097

1103

1109

1117

1123

1129

1151

1153

1163

1171

1181

1187

1193

1201

1213

1217

1223

1229

1231

1237

1249

1259

1277

1279

1283

1289

1291

1297

1301

1303

1307

1319

1321

1327

1361

1367

1373

1381

1399

1409

1423

1427

1429

1433

1439

1447

1451

1453

1459

1471

1481

1483

1487

1489

1493

1499

1511

1523

1531

1543

1549

1553

1559

1567

1571

1579

1583

1597

1601

1607

1609

1613

1619

1621

1627

1637

1657

1663

1667

1669

1693

1697

1699

1709

1721

1723

1733

1741

1747

1753

1759

1777

1783

1787

1789

1801

1811

1823

1831

1847

1861

1867

1871

1873

1877

1879

1889

1901

1907

1913

1931

1933

1949

1951

1973

1979

1987

1993

1997

1999

2003

2011

2017

2027

2029

2039

2053

2063

2069

2081

2083

2087

2089

2099

2111

2113

2129

2131

2137

2141

2143

2153

2161

2179

2203

2207

2213

2221

2237

2239

2243

2251

2267

2269

2273

2281

2287

2293

2297

2309

2311

2333

2339

2341

2347

2351

2357

2371

2377

2381

2383

2389

2393

2399

2411

2417

2423

2437

2441

2447

2459

2467

2473

2477

2503

2521

2531

2539

2543

2549

2551

2557

2579

2591

2593

2609

2617

2621

2633

2647

2657

2659

2663

2671

2677

2683

2687

2689

2693

2699

2707

2711

2713

2719

2729

2731

2741

2749

2753

2767

2777

2789

2791

2797

2801

2803

2819

2833

2837

2843

2851

2857

2861

2879

2887

2897

2903

2909

2917

2927

2939

2953

2957

2963

2969

2971

2999

3001

3011

3019

3023

3037

3041

3049

3061

3067

3079

3083

3089

3109

3119

3121

3137

3163

3167

3169

3181

3187

3191

3203

3209

3217

3221

3229

3251

3253

3257

3259

3271

3299

3301

3307

3313

3319

3323

3329

3331

3343

3347

3359

3361

3371

3373

3389

3391

3407

3413

3433

3449

5003

5009

5011

5021

5023

5039

5051

5059

5077

5081

5087

5099

5101

5107

5113

5119

5147

5153

5167

5171

10007

10009

10037

10039

10061

10067

10069

10079

10091

10093

10099

10103

10111

10133

10139

10141

10151

10159

10163

10169

20011

20021

20023

20029

20047

20051

20063

20071

20089

20101

20107

20113

20117

20123

20129

20143

20147

20149

20161

20173

40009

40013

40031

40037

40039

40063

40087

40093

40099

40111

40123

40127

40129

40151

40153

40163

40169

40177

40189

40193

80021

80039

80051

80071

80077

80107

80111

80141

80147

80149

80153

80167

80173

80177

80191

80207

80209

80221

80231

80233

160001

160009

160019

160031

160033

160049

160073

160079

160081

160087

160091

160093

160117

160141

160159

160163

160169

160183

160201

160207

320009

320011

320027

320039

320041

320053

320057

320063

320081

320083

320101

320107

320113

320119

320141

320143

320149

320153

320179

320209

640007

640009

640019

640027

640039

640043

640049

640061

640069

640099

640109

640121

640127

640139

640151

640153

640163

640193

640219

640223

1000003

1000033

1000037

1000039

1000081

1000099

1000117

1000121

1000133

1000151

1000159

1000171

1000183

1000187

1000193

1000199

1000211

1000213

1000231

1000249

Notes

The Table contains all primes through 3449, and selected ranges of larger primes.

The linear system being analyzed comprises 324 equations in 448 variables. Its rank is 282.

It's not surprising that for the small prime numbers illustrated in the table the computed dimension is not always correct.

On the other hand, note that the dimension is often computed correctly even though the residuals suggest that there is a problem.

For large prime numbers the residuals are all non-zero, as one would expect.

It is remarkable that the primes for which the dimension is overestimated occur in blocks.